Coherent potential approximation

The first scheme I consider is sometimes called the Coherent Potential Approximation (CPA) (Gubernatis and Krumhansl, 1975; Berryman, 1992; Berryman and Berge, 1996) or the Self-Consistent Scheme (Korringa et al., 1979; Berryman, 1980).

When there is no pore fluid present (i.e., drained frame conditions), the equations of poroelasticity reduce to those of elasticity (general) for the porous frame material. Within CPA, the idea is to treat all constituents on an equal footing, so no single material serves as host medium for the others. For this reason, the CPA is sometimes known as a symmetrical self-consistent scheme. To find the formulas for the CPA, we take the reference material to be the composite itself, so r=*. The formula (general) reduces to

v_i (C

^(i)-C

^*_CPA)^*i = 0,   where I have now approximated the unknown linear coefficient by the Eshelby-Wu tensor $\matT^{*i}$ corresponding to inclusions of stiffness ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}^{(i)}$ in host material of stiffness ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}^*_{CPA}$.

To make use of the generalization of Eshelby's formula for poroelasticity in the case when pore fluid and pore pressure are significant factors, I note that each inclusion is effectively imbedded in the composite material *, so it makes sense to consider the formula for inclusion strain

^(i) = e^*i(p_f) + ^*i[^* - e^*i(p_f)],   where the strain corresponding to equal expansion or contraction of both materials i and * is given by

e^*i_pq = (^*-^(i)K^*-K^(i))p_f3_pq.   If the mixture were composed only of the two materials i and *, then the uniform expansion result would apply exactly. In the composite poroelastic material, (CPAEshelby) should be viewed as an estimate of the true strain of the ith constituent. This estimate is conceptually on the same footing as that traditionally used when saying that $\varepsilon^{(i)} = \matT^{*i}\varepsilon^*$ is a reasonable approximation of the strain in the ith constituent of an elastic composite, even though there may be many other types of materials present.

To derive a formula within CPA for the Biot-Willis constant $\alpha^*$,I want to make use of (CPAEshelby). For elasticity, the average confining stress equals the total confining stress, so $\sum v_i\sigma_i = \sigma$. This fact was actually used to derive (general). However, for poroelasticity with finite pore pressure p_f, it is important to distinguish confining stress from the stress in the solid components and so it is no longer true that the average confining stress is equal to the total confining stress, i.e., $\sum v_i \sigma^{(i)} \ne \sigma$. The correct relation for the pertinent stress is more complicated than this. (We could learn some important things about our problem by studying this issue, but the analysis becomes rather more technical than what follows and it seems preferable to avoid this discussion here.) However, it is still necessarily true that the average strain is equal to the total strain, i.e.,

v_i ^(i) = ^*.   Furthermore, this relation is just what is needed to make use of (CPAEshelby). Substituting (CPAEshelby) into (averagestrain) and then rearranging terms, I find that

v_i (- ^*i) e^*i(p_f) = v_i (- ^*i)^*,   where is the identity matrix and I used the fact that $\sum v_i = 1$.Equation (CPAalpha) is almost what I want, but the right hand side seems to be a problem, because it depends explicitly on $\varepsilon^{*}$, which is arbitrary. It is known however that $\sum v_i(\matI - \matG^{*i}) \equiv 0$(Hill, 1963; Berryman and Berge, 1996), and, since $\matT^{*i}$ is my approximation to $\matG^{*i}$, it is clear that the right hand side of (CPAalpha) should be set identically to zero. Thus, after making use of (CPAstrain) in (CPAalpha), the CPA for $\alpha^*$ is

v_i (1 - P^*i)^*_CPA-^(i)K^*_CPA - K^(i) = 0,   where P^*i is the coefficient for the compressional modulus, and the corresponding coefficient for the shear modulus is Q^*i (see Table 1). Some care should be taken however to check the degree of satisfaction of the subsidiary condition $\sum v_i(\matI - \matT^{*i}) \simeq 0$ to make sure that it is at least approximately satisfied by the estimate obtained for ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}^*_{CPA}$. It turns out that this condition is satisfied exactly for spherical inclusions. Furthermore, in the case of spheres I have $P^{*i} = (K^*+{4\over3}\mu^*)/(K^{(i)}+{4\over3}\mu^*)$,and it is easy to show that (CPAalpha) reduces to

v_i (^(i)-^*_CPA)P^*i = 0,   which should be compared to

v_i (K^(i)-K_CPA^*)P^*i = 0,   which follows from (CPA).

TABLE 1. Four examples of coefficients P and Q for spherical and nonspherical scatterers. The superscripts h and i refer to host and inclusion phases, respectively. Special characters are defined by $\beta= \mu[(3K+\mu)/(3K+4\mu)]$,$\gamma= \mu[(3K+\mu)/(3K+7\mu)]$, and $\zeta= (\mu/6)[(9K+8\mu)/(K+2\mu)]$.The expression for spheres, needles, and disks were derived by Wu (1966) and Walpole (1969). The expressions for penny-shaped cracks were derived by Walsh (1969) and assume K⁽ⁱ⁾/K^(h) << 1 and $\mu^{(i)}/\mu^{(h)} << 1$.The aspect ratio of the cracks is a.